| Unit 1 |
Patterns & Equations - Main Ideas |
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- Patterns can be represented in a variety of ways using words, numbers, symbols, tables and graphs
- The same pattern can be represented in different ways.
- Patterns can be used to solve problems.
- An equation is a statement that two expressions have the same value.
- If both sides of an equation are changed in the same way, the equation remains balanced.
- Addition and multiplication are commutative operations.
- [BACK]
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| L-1 |
- A pattern rule can be illustrated with an Input/Output machine.
- The input and output can be recorded in a table. The input and output can be described using pattern rules. There is a relationship between corresponding input numbers and output numbers.
- [BACK]
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| L-2 |
- A pattern in a table of values can be represented pictorially, concretely, and numerically
- A pattern rule describes the relationship between the two columns in a table of values.
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| L-3 |
- Some problems can be solved using patterns and setting up tables.
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| L-4 |
- A variable in an expression can be used to represent a pattern rule.
- An expression with a variable can be used to solve some problems.
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| L-5 |
- Ordered pairs are used to plot or locate points on a coordinate grid.
- In an ordered pair, the first number tells the horizontal distance from the origin; the second number tells the vertical distance from the origin.
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| L-6 |
- A graph can be drawn to represent the pattern in an input/output table.
- The numbers in a table of values represent the ordered pairs on a graph.
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| L-7 |
- The equals sign shows that the expressions on both sides of an equation are equal.
- Addition and multiplication are commutative. The order in which two numbers are added or multiplied does not affect the sum or product.
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| L-8 |
- When each side of an equation is changed in the same way, the values remain equal. This is called preservation of equality.
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| Unit 2 |
Understanding Number - Main Ideas |
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- The position of a digit in a number determines its value.
- There are patterns in the way numbers are formed.
- Large numbers are best understood in terms of familiar real-world referents.
- Every number has both size and a positive or negative relationship to other numbers.
- Estimation plays an important role in all computations.
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| L-1 |
- Large numbers are organized into groups of three place values. Each group is called a period.
- Within each period, the digits are read as hundreds, tens, and ones.
- When we read large numbers, we say the period name after each period except the units period.
- When we write a number with five or more digits, we leave a space between the periods
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| L-2 |
- Part of developing a good understanding of computation is being able to select the most appropriate method of computing for the problem at hand.
- When a precise answer is required, a calculator is used for more complex calculations. Estimation is used to judge the reasonableness of answers.
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| L-3 |
- The multiples of a number can be found by starting at the number of counting on by that number.
- A common multiple is a number that is a multiple of each of two or more given numbers.
- The least common multiple is the first common multiple of two or more given numbers.
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| L-4 |
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| L-5 |
- A number that is a factor of each of two or more different numbers is a common factor.
- The factors of a number can be sorted into factors that are composite and factors that are prime.
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| L-6 |
- Making an organized list is a good strategy for solving a problem in which more than one pair of numbers must be tried.
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| L-7 |
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| L-8 |
- Integers greater than zero are positive integers.
- Integers less than zero are negative integers.
- Opposite integers are the same distance from zero, but on opposite sides of zero.
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| L-9 |
- Greater integers are to the right of lesser integers on a number line.
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| Unit 3 |
Decimals - Main Ideas |
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- The place-value chart extends infinitely to the right to display numbers with decimal places beyond thousandths. Each position in the chart represents 10 times as many as the position to its right.
- Regardless of the positions of the decimal points, the multiplication or division of two numbers will produce the same digits. For example, 2 X 3 = 6 and 0.2 X 3 = 0.6 both have 6 in their products, with the one difference being the placement of the decimal point. Thus, the computations can be performed as whole numbers, and the placement of the decimal point can be determined by estimation.
- Estimation plays an important role in decimal computation.
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| L-1 |
- Patterns in the place-value chart help us to know the position and value of each digit in a decimal number
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| L-2 |
- Strategies such as front-end estimation, decimal benchmarks, and compatible numbers can be used to estimate products and quotients with decimals.
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| L-3 |
- Multiplying a decimal by a whole number is like multiplying two whole numbers.
- Estimation can be used to place the decimal point in the product.
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| L-4 |
- When you multiply a decimal less than 1 by a whole number, the product is less than the whole number.
- Place value and estimation can be used to multiply a decimal less than 1 by a whole number.
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| L-5 |
- Dividing a decimal by a whole number involves the same procedures as dividing a whole number by a whole number.
- Estimation can be used to place the decimal point.
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| L-6 |
- To reach a remainder of zero when dividing a decimal by a whole number, it is sometimes necessary to add zeros in the dividend.
- Sometimes you may never reach a remainder of zero, no matter how many zeros you write in the dividend. Then the quotient is approximate.
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| L-7 |
- When you divide a decimal less than 1 by a whole number, the quotient is less than both the dividend and the divisor.
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| L-8 |
- Understanding what a problem is about is an important step toward solving the problem.
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| Unit 4 |
Angles and Polygons - Main Ideas |
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- Angles can be compared, measured, and constructed using non-standard measuring tools and standard protractors.
- Angles are measured in degrees.
- Angles can be classified according to their measures.
- Generalizations can be made about the sums of interior angles in triangles and quadrilaterals.
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| L-1 |
- An angle is formed when two lines meet.
- The size of an angle is a measure of the amount of turn needed to move from one arm to the other.
- An angle can be named by the way it relates to a right angle or a straight angle.
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| L-2 |
- Concrete, non-standard units can be used to measure and compare angles.
- Protractors marked with non-standard units can be used to measure and compare angles.
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| L-3 |
- Angles can be measured using a standard protractor.
- Angles are classified according to their measures in degree.
- To estimate the measure of an angle, we can use 45 degrees, 90 degrees, and 180 degrees as reference angles.
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| L-4 |
- An angle of a given measure can be constructed using a standard protractor.
- Angles of 45º, 90º, and 180º can be drawn without using a protractor.
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| L-5 |
- Check and reflect is an important step in solving problems. It helps to ensure the accuracy and reasonableness of solutions.
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| L-6 |
- The sum of the interior angles in any triangle is 180º
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| L-7 |
- The sum of the angles in a quadrilateral is 360º
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| Unit 5 |
Fractions, Ratios, and Percents - Main Ideas |
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- Fractions that represent more than 1 can be written as mixed numbers or improper fractions
- A ratio is a comparison of any two quantities. Equivalent ratios can be found by using concrete materials or a chart and patterns.
- A percent is a special ratio that compares a number to 100. Thus, percent means the number of hundredths. Percents are another way to write fractions and decimals.
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| L-1 |
- An improper fraction represents more than one whole
- A mixed number, with a whole number part and a fraction part, is another way of writing an improper fractions
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| L-2 |
- Both improper fractions and mixed numbers represent more than one whole.
- A mixed number can be converted into an improper fraction.
- An improper fraction can be converted into a mixed number
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| L-3 |
- A number line can be used to compare mixed numbers and improper fractions.
- Mixed numbers and improper fractions can be compared by converting the mixed number to an improper fraction, then writing each improper fraction as an equivalent fraction with like denominators.
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| L-4 |
- A ratio is a comparison of two quantities with the same units.
- The order of the terms of a ratios is important.
- A ratio can be used to compare a part of a set to another part of the set.
- A ratio can be used to compare a part of a set to the whole set.
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| L-5 |
- Equal sets of ratios are called equivalent ratios.
- You can find equivalent ratios by multiplying or dividing the terms of a ratio by the same non-zero number.
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| L-6 |
- When solving a problem, thinking can be represented with words, diagrams, and/or numbers.
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| L-7 |
- Percent means “per hundred” or “out of 100.”
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| L-8 |
- Fractions, decimals, and percents are three ways to describe parts of one whole.
- You can use a percent to describe any part of one whole by writing an equivalent fraction with hundredths.
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| Unit 6 |
Geometry and Measurement - Main Ideas |
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- Shapes can be classified by attributes, such as side length, angle measure, and number of equal sides.
- Triangles can be named and sorted by types of angles and by number of equal sides.
- A regular polygon has all sides equal and all angles equal. A regular polygon also has line symmetry.
- Two shapes that have the same size and shape, but not necessarily the same orientation, are congruent.
- Formulas can be used to determine the perimeters of polygons, the area of rectangles, and the volume of right rectangular prisms.
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| L-1 |
- We can name triangles according to the number of equal sides.
- We can name triangles according to the number of equal angles.
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| L-2 |
- We can name triangles according to the types of interior angles.
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| L-3 |
- Triangles can be constructed using a ruler and protractor, given side and angle measures.
- Two triangles can have congruent angles but different side lengths.
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| L-4 |
- A regular polygon has all sides equal, all angles equal, and line symmetry. An irregular polygon does not have all sides equal and all angles equal.
- A convex polygon has all angles less than 180º. A concave polygon has at least one angle greater than 180º.
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| L-5 |
- When polygons match exactly, the polygons are congruent.
- Congruent polygons have all sides equal and all angles equal.
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| L-6 |
- Solving a simpler problem and extending a table are two effective strategies to solve problems.
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| L-7 |
- The perimeter of a polygon can be found by adding the side lengths.
- Formulas can be used to calculate the perimeters of specific polygons.
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| L-8 |
- The area of a rectangle can be found by using the formula A = L X W, where A represents the area of the rectangle, L represents its length, and W represents its width.
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| L-9 |
- The volume of an object is a measure of the space it takes up.
- The volume of a rectangular prism can be found using a formula.
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| Unit 7 |
Data Analysis & Probability - Main Ideas |
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- Data can be collected, organized, and displayed in a variety of ways.
- Well-designed questionnaires help us collect valid data.
- Line graphs are appropriate for displaying continuous rather than discrete data.
- Graphs help us draw conclusions about data.
- Probability means the likelihood of an event.
- When all outcomes are equally likely, theoretical probability is calculated by comparing the number of favourable outcomes to the number of possible outcomes.
- Experimental probability is calculated by comparing the number of times an outcome occurs to the number of times the experiment is conducted.
- Experimental probabilities often differ from theoretical probabilities. The more times an experiment is conducted, the closer the experimental probability may be to the theoretical probability.
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| L-1 |
- A questionnaire can be used to collect data.
- A question should be understood in the same way by all, contain a possible answer for all, and not influence a person to answer in a certain way.
- A question that might persuade a person to answer in a certain way is biased.
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| L-2 |
- Data can be gathered by recording the results of experiments.
- Data from experiments can be used to draw conclusions.
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| L-3 |
- Discrete data represent things that can be counted. Continuous data can include any value between data points. Time, money, temperature, and measurements, such as length and mass, are continuous.
- A line graph shows continuous data. Consecutive points are joined by line segments.
- A line graph is not appropriate for discrete data. the graph s a series of points that are not joined.
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| L-4 |
- Graphs can be used to display both continuous and discrete data.
- When the data are continuous, we join consecutive points with line segments.
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| L-5 |
- Different sets of data may require different types of graphs
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| L-6 |
- Theoretical probability is the likelihood that an outcome will happen.
- When all outcomes are equally likely, theoretical probability is calculated by dividing the number of favourable outcomes by the number of possible outcomes.
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| L-7 |
- Experimental probabilities are based on the results of an experiment.
- Experimental probabilities can only be calculated for experiments with outcomes that are equally likely.
- The more times an experiment is repeated, the closer the experimental probability may come to the theoretical probability
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| L-8 |
- There are a variety of ways to explain mathematical thinking.
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| Unit 8 |
Transformations - Main Ideas |
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- Coordinate systems can be used to describe the location of a shape
- Coordinate systems can be used to describe transformations of shapes.
- A shape and its transformation images are congruent.
- Often, more than one combination of transformations is possible to move a shape to its final image.
- Transformations of one or more shapes can be used to create a design.
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| L-1 |
- Ordered pairs can be used to describe the position of a shape on a Cartesian plane.
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| L-2 |
- The coordinates of the vertices can be used to describe a transformation image on a coordinate grid.
- When points on a transformation image are labelled, an apostrophe after the label refers to the corresponding image point. For example, A’ is the image of point A.
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| L-3 |
- The same transformation can be applied to a shape more than once.
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| L-4 |
- A combination of 2 or 3 different types of transformations can be applied to a shape on a grid.
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| L-5 |
- Transformations of one or more shapes can be used to create a design.
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| L-6 |
- Guess and test is a good strategy to use when solving a problem with more than one possible solution.
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